Abstract
We give a proof of the Lieb-Thirring inequality in the critical case d=1, γ = 1/2, which yields the best possible constant.
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M. Aizenmann and E. H. Lieb: On semi-classical bounds for eigenvalues of Schrödinger operators. Phys. Lett. 66A (1978), 427–429.
M. S. Birman: The spectrum of singular boundary problems. Mat. Sb. 55 No. 2 (1961), 125–174, translated in Amer. Math. Soc. Trans. (2), 53 (1966), 23-80.
J. G. Conlon: A new proof of the Cwikel-Lieb-Rosenbljum bound. Rocky Mountain J. Math., 15, no.1 (1985), 117–122.
M. Cwikel: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Trans. AMS, 224 (1977), 93–100.
L. D. Landau and E. M. Lifshitz: Quantum Mechanics. Non-relativistic theory. Volume 3 of Course of Theoretical Physics, Pergamon Press (1958)
P. Li and S.-T. Yau: On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys., 88 (1983), 309–318.
E. H. Lieb: The number of bound states of one body Schrödinger operators and the Weyl problem. Bull. Amer. Math. Soc., 82 (1976), 751–753. See also Proc. A.M.S. Symp. Pure Math. 36 (1980), 241-252.
E. H. Lieb and M. Loss: Analysis. Graduate Studies in Mathematics 14, American Mathematical Society 1997.
E. H. Lieb and W. Thirring: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett., 35 (1975), 687–689. Errata 35 (1975), 1116.
E. H. Lieb and W. Thirring: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Studies in Math. Phys., Essays in Honor of Valentine Bargmann, Princeton (1976)
G. V. Rozenbljum: Distribution of the discrete spectrum of singular differential operators. Dokl. AN SSSR, 202, N 5 1012–1015 (1972), Izv. VUZov, Matematika, N. 1(1976), 75-86.
M. Reed and B. Simon: Methods of modern mathematical physics IV: Analysis of operators. Academic Press, New York 1978.
J. Schwinger: On the bound states of a given potential. Proc. Nat. Acad. Sci. U.S.A. 47, (1961), 122–129.
B. Simon: Quantum mechanics for Hamiltonians defined as quadratic forms. Princeton Series in Physics, Princeton University press, New Jersey, 1971.
B. Simon: The bound state of weakly coupled Schrdinger operators in one and two dimensions. Ann. Physics 97, no. 2, (1976), 279–288.
W. Thirring: A course in mathematical physics. Vol. 3. Quantum mechanics of atoms and molecules. Translated from the German by Evans M. Harrell. Lecture Notes in Physics, 141. Springer-Verlag, New York-Vienna, 1981.
T. Weidl: On the Lieb-Thirring constants L γ 1 for γ ≥ 1/2. Comm. Math. Phys., 178, no. 1, (1996), 135–146.
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Hundertmark, D., Lieb, E.H., Thomas, L.E. (2002). A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator. In: Loss, M., Ruskai, M.B. (eds) Inequalities. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55925-9_28
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DOI: https://doi.org/10.1007/978-3-642-55925-9_28
Publisher Name: Springer, Berlin, Heidelberg
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